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Chapter 3: Problem 7

a. A consumer is willing to trade 3 units of \(x\) for 1 unit of \(y\) when she has 6 units of \(x\) and 5 units of \(y\). She is also willing to trade in 6 units of \(x\) for 2 units of \(y\) when she has 12 units of \(x\) and 3 units of \(y .\) She is indifferent between bundle (6,5) and bundle \((12,3) .\) What is the utility function for goods \(x\) and \(y^{3}\) Hint: What is the shape of the indifference curve? b. A consumer is willing to trade 4 units of \(x\) for 1 unit of \(y\) when she is consuming bundle \((8,1) .\) She is also willing to trade in 1 unit of \(x\) for 2 units of \(y\) when she is consuming bundle (4,4) . She is indifferent between these two bundles. Assuming that the utility function is Cobb-Douglas of the form \(U(x, y)=x^{2} y^{3},\) where \(\alpha\) and \(\beta\) are positive constants, what is the utility function for this consumer? c. Was there a redundancy of information in part (b)? If yes, how much is the minimum amount of information required in that question to derive the utility function?

Short Answer

Expert verified
Was there a redundancy of information in part B? Answer: The utility function for the consumer in part A is U(x, y) = x * y, while in part B, the utility function is U(x, y) = x^2 * y^3. There was a redundancy of information in part B, as only one of the trading ratios was needed to derive the utility function.

Step by step solution

01

Find the marginal rate of substitution (MRS) for each trading scenario

Given the trading ratios, the MRS is the rate at which the consumer is willing to trade good x for good y. In the first scenario, the consumer is willing to trade 3 units of x for 1 unit of y. Therefore, MRS1 is 3. In the second scenario, the consumer is willing to trade 6 units of x for 2 units of y. Therefore, MRS2 is 3 (6/2 = 3). The MRS for both scenarios is the same, which tells us that the indifference curve has a constant slope.
02

Derive the utility function

Since the indifference curve has a constant slope, we know that the utility function is of the form U(x, y) = x^α * y^β. The consumer is indifferent between bundles (6,5) and (12,3), so their utility function must satisfy: U(6,5) = U(12,3) With a constant MRS, we can be sure that α = β (assuming constant returns to scale). Let’s assume that the utility function is of the form U(x, y) = x^α * y^α, then: (6^α * 5^α) = (12^α * 3^α) By solving for α, we get: α = 1 Thus, the utility function for goods x and y is U(x, y) = x * y. #Part B#
03

Write down the utility function

Since the utility function is given in a Cobb-Douglas form, we can write it down as: U(x, y) = x^α * y^β
04

Derive the MRS for the utility function

Using the utility function U(x, y) = x^α * y^β, we have that the MRS is equal to: MRS = - (α * x^(α - 1) * y^β) / (β * x^α * y^(β - 1)) By simplifying the expression, we obtain: MRS = - (α / β) * (y / x)
05

Use the given trading ratios to solve for α and β

The consumer is willing to trade 4 units of x for 1 unit of y when consuming bundle (8,1). With the given MRS, we have: - (α / β) * (1 / 8) = 4 The consumer is also willing to trade 1 unit of x for 2 units of y when consuming bundle (4,4). Using the MRS, we have: - (α / β) * (4 / 4) = 1 / 2 By solving these simultaneous equations for α and β, we get: α = 2 β = 3
06

State the utility function

Since α = 2 and β = 3, we can now write the utility function for this consumer as: U(x, y) = x^2 * y^3 #Part C#
07

Analyze the redundancy of information in part (b)

In part (b), we found that α = 2 and β = 3 by using the two given trading ratios in conjunction with the MRS. However, given that we already know the Cobb-Douglas form of the utility function, we only need one of the trading ratios to derive the utility function.
08

Determine the minimum amount of information required

Since we only need one of the given trading ratios to derive the utility function, there is a redundancy of information in part (b). The minimum amount of information required is one of the trading ratios (either 4 units of x for 1 unit of y when consuming bundle (8,1), or 1 unit of x for 2 units of y when consuming bundle (4,4)).

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Most popular questions from this chapter

The formal study of preferences uses a general vector notation. A bundle of \(n\) commoditics is denoted by the vector \(\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right),\) and a preference relation \((>)\) is defined over all potential bundles. The statement \(\mathbf{x}^{1}>\mathbf{x}^{2}\) means that bundle \(\mathbf{x}^{1}\) is preferred to bundle \(\mathbf{x}^{2}\). Indifference between two such bundles is denoted by \(\mathbf{x}^{1} \approx \mathbf{x}^{2}\) The preference relation is "complete" if for any two bundles the individual is able to state cither \(\mathbf{x}^{1}>\mathbf{x}^{2}, \mathbf{x}^{2}>\mathbf{x}^{1}\), or \(\mathbf{x}^{1} \approx\) \(\mathbf{x}^{2}\). The relation is "transitive" if \(\mathbf{x}^{1}>\mathbf{x}^{2}\) and \(\mathbf{x}^{2}>\mathbf{x}^{3}\) implies that \(\mathbf{x}^{1}>\mathbf{x}^{3}\), Finally, a preference relation is "continuous" if for any bundle \(y\) such that \(y>x\), any bundle suitably close to \(y\) will also be preferred to \(x\). Using these definitions, discuss whether each of the following preference relations is complete, transitive, and continuous. a Summation preferences: This preference relation assumes one can indeed add apples and oranges. Specifically, \(\mathbf{x}^{1}>\mathbf{x}^{2}\) if and only if \(\sum_{i=1}^{n} x_{i}^{1}>\sum_{i=1}^{n} x_{i}^{2} .\) If \(\sum_{i=1}^{n} x_{i}^{1}=\sum_{i=1}^{n} x_{i}^{2}, \mathbf{x}^{1} \approx \mathbf{x}^{2}\) b. Lexicographic preferences: In this case the preference relation is organized as a dictionary: If \(x_{1}^{1}>x_{1}^{2}, x^{1} \succ x^{2}\) (regardless of the amounts of the other \(n-1\) goods). If \(x_{1}^{1}=x_{1}^{2}\) and \(x_{2}^{1}>x_{2}^{2}, x^{1}>x^{2}\) (regardless of the amounts of the other \(n-2\) goods). The lexicographic preference relation then continues in this way throughout the entire list of goods. c. Preferences with satiation: For this preference relation there is assumed to be a consumption bundle (x") that provides complete "bliss." The ranking of all other bundles is determined by how close they are to \(\mathbf{x}^{*}\), That is, \(\mathbf{x}^{1}>\mathbf{x}^{2}\) if and only if \\[ \left|\mathbf{x}^{1}-\mathbf{x}^{*}\right|<\left|\mathbf{x}^{2}-\mathbf{x}^{\prime}\right| \text { where }\left|\mathbf{x}^{l}-\mathbf{x}^{*}\right|=\sqrt{\left(x_{1}^{\prime}-x_{1}^{*}\right)^{2}+\left(x_{2}^{\prime}-x_{x}^{*}\right)^{2}+\ldots+\left(x_{n}^{\prime}-x_{n}^{*}\right)^{2}} \\]

Find utility functions given each of the following indifference curves [defined by \(U(')=k]\) a \(z=\frac{k^{1 / 8}}{x^{a / b} y^{4 / 8}}\) b. \(y=0.5 \sqrt{x^{2}-4\left(x^{2}-k\right)}-0.5 x\) \(c_{1} z=\frac{\sqrt{y^{4}-4 x\left(x^{2} y-k\right)}}{2 x}-\frac{y^{2}}{2 x}\)

As we saw in Figure \(3.5,\) one way to show convexity of indifference curves is to show that, for any two points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) on an indifference curve that promises \(U=k\), the utility associated with the point \(\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)\) is at least as great as \(k\). Use this approach to discuss the convexity of the indifference curves for the following three functions. Be sure to graph your results. a. \(U(x, y)=\min (x, y)\) b. \(U(x, y)=\max (x, y)\) c. \(U(x, y)=x+y\)

Consider the following utility functions: a \(U(x, y)=x y\) \(U(x, y)=x^{2} y^{2}\) \(c(x, y)=\ln x+\ln y\) Show that each of these has a diminishing \(M R S\) but that they exhibit constant, increasing, and decreasing marginal utility, respectively. What do you conclude?

Graph a typical indifference curve for the following utility functions, and determine whether they have convex indifference curves (i.e., whether the \(M R S\) declines as \(x\) increases). a. \(U(x, y)=3 x+y\) b. \(U(x, y)-\sqrt{x \cdot y}\) c. \(U(x, y)=\sqrt{x}+y\) \(\mathrm{d} U(x, y)=\sqrt{x^{2}-y^{2}}\) e. \(U(x, y)=\frac{x y}{x+y}\)
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